Equivariant homotopy theory and gauge theory

在Seiberg-Witten理论中，M.Furuta引入了一个有限维近似来捕捉等变同伦理论中的方程，并由此得到了闭自旋4-流形的不变量和10/8-不等式的一个精化.在本研究中，我们考虑了一维上同调上的四重结构，从而改进了这个不等式.更确切地说，我们定义了闭自旋4-流形f的KO特征的一个变体，并得到了由这个不变量决定的一个附加项。我们还证明了，如果四元环的结构与四维环的环或其副本的连通和模2同余，则我们的改进是可以估计的。据M.Furuta报道，他正在应用这一结果来研究辛4-流形的Seiberg-Witten不变量。在完成这项工作后，我们考虑了我们的结果与方程解的模空间的几何的关系。这最初是由P.Kroneheimer研究的，他考虑了低维模空间的这一点。为了将他的方法推广到高维模空间，我们在模空间上引入了一种KO-特征。然后，利用模空间的对称性及其自旋结构，可以直接得到10/8-不等式以及上述改进。现在，我们正在尝试将这种方法应用于其他情况，如杨-米尔斯理论。

In Seiberg-Witten theory M.Furuta has introduced a finite dimensional approximation to capture the equation in equivariant homotopy theory, by which he has obtained a refinement of the invariant and the 10/8-inequality for closed spin 4-manifolds.In this research we improved this inequality by taking into account the quadruple structure on one dimensional cohomology. More precisely we defined a variant of KO-characteristics for closed spin 4-manifoldfs and obtained an additional term determined by this invariant. We also showed that, if the quadruple structure is congruent to the one of 4-dimesional torus or the connected sum of its copies modulo 2, our improvement can be estimated. The researcher was reported by M.Furuta that he is now applying this result to study Seiberg-Witten invariants for symplectic 4-manifolds.After finishing this work we considered how our result is related to geometry of the moduli space of solutions to the equation. Originally this was studied by P.Kroneheimer, who considered this for low-dimensional moduli spaces. To extend his method to higher dimensional moduli spaces, we introduced a sort of KO-characteristics on the moduli space. Then the 10/8-inequality, as well as the above improvement, can be directly obtained from symmetry of the moduli space with its spin structure. Now we are trying to apply this method in other situations as Yang-Mills theory.